We need to solve the compound inequality:
\[ -5 \leq \frac{k - 10}{3} < -2 \]
To eliminate the fraction, we can multiply all parts of the inequality by 3. However, we must remember that since 3 is positive, we do not change the direction of the inequalities:
\[ -5 \cdot 3 \leq k - 10 < -2 \cdot 3 \]
This gives us:
\[ -15 \leq k - 10 < -6 \]
Next, we will add 10 to all parts of the compound inequality to isolate \( k \):
\[ -15 + 10 \leq k < -6 + 10 \]
Simplifying the inequalities, we get:
\[ -5 \leq k < 4 \]
Now, writing the answer as a compound inequality with integers, we have:
\[ k \in [-5, 4) \]
So the solution in compound inequality form is:
\[ -5 \leq k < 4 \]